Simplify the following expression and state the condition under which the simplification is valid: $n = \dfrac{r^2 + 7r}{r^2 + 14r + 49}$
First factor the expressions in the numerator and denominator. $ \dfrac{r^2 + 7r}{r^2 + 14r + 49} = \dfrac{(r)(r + 7)}{(r + 7)(r + 7)} $ Notice that the term $(r + 7)$ appears in both the numerator and denominator. Dividing both the numerator and denominator by $(r + 7)$ gives: $n = \dfrac{r}{r + 7}$ Since we divided by $(r + 7)$, $r \neq -7$. $n = \dfrac{r}{r + 7}; \space r \neq -7$